Sketch the Graph of f(x) = 2 cosec πx
Question:
Sketch the graph of the following function :
\[ f(x)=2\cosec \pi x \]
Solution:
We know that
\[ \cosec \theta=\frac{1}{\sin\theta} \]
Therefore
\[ f(x)=2\cosec \pi x=\frac{2}{\sin \pi x} \]
The graph of cosecant is obtained from the graph of sine.
Whenever
\[ \sin \pi x=0 \]
the function becomes undefined.
Thus vertical asymptotes occur at
\[ x=0,\ 1,\ 2,\ 3,\dots \]
Important properties:
- Period \(=\dfrac{2\pi}{\pi}=2\)
- Range \(y\le -2\) or \(y\ge 2\)
- Vertical asymptotes at integral values of \(x\)
Now calculate some important points:
\[ \begin{aligned} x=\frac12 &\Rightarrow y=2\cosec\frac{\pi}{2}=2\\[8pt] x=\frac32 &\Rightarrow y=2\cosec\frac{3\pi}{2}=-2\\[8pt] x=\frac52 &\Rightarrow y=2\cosec\frac{5\pi}{2}=2 \end{aligned} \]
Thus the graph passes through the points
\[ \left(\frac12,2\right),\quad \left(\frac32,-2\right),\quad \left(\frac52,2\right) \]
Plot these points and draw the cosecant curves approaching the vertical asymptotes.
Hence, the required graph is shown above.